A180332 Primitive Zumkeller numbers.

6, 20, 28, 70, 88, 104, 272, 304, 368, 464, 496, 550, 572, 650, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030, 4070, 4095, 4216

Offset: 1

Author

T. D. Noe, Sep 07 2010

Keywords

• nonn

Comments

A number is called a primitive Zumkeller number if it is a Zumkeller number (A083207) but none of its proper divisors are Zumkeller numbers. These numbers are very similar to primitive non-deficient numbers (A006039), but neither is a subsequence of the other. [See A378538, A378656, A378657].

Because every Zumkeller number has a divisor that is a primitive Zumkeller number, every Zumkeller number z can be factored as z = d*r, where d is the smallest divisor of z that is a primitive Zumkeller number.

Every number of the form p*2^k is a primitive Zumkeller number, where p is an odd prime and k = floor(log_2(p)).

The odd terms are not the same as A006038. For example, 342225 occurs there, but not here, while 4448925 occurs here, but is not in A006038. - Antti Karttunen, Dec 05 2024

Links

• T. D. Noe, Table of n, a(n) for n = 1..9179
• Index entries for sequences where odd perfect numbers must occur, if they exist at all

Crossrefs

Cf. A000396 (subsequence), A006038, A083207, A006039, A378537 (characteristic function), A378538, A378656, A378657.

Programs

Mathematica

ZumkellerQ[n_] := ZumkellerQ[n] = Module[{d = Divisors[n], ds, x}, ds = Total[d]; If[OddQ[ds], False, SeriesCoefficient[Product[1 + x^i, {i, d}], {x, 0, ds/2}] > 0]];
Reap[For[n = 1, n <= 5000, n++, If[ZumkellerQ[n] && NoneTrue[Most[Divisors[ n]], ZumkellerQ], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 01 2019 *)

Python

from sympy import divisors
from sympy.utilities.iterables import subsets
def isz(n): # after Peter Luschny in A083207
    divs = divisors(n)
    s = sum(divs)
    if not (s%2 == 0 and 2*n <= s): return False
    S = s//2 - n
    R = [m for m in divs if m <= S]
    return any(sum(c) == S for c in subsets(R))
def ok(n): return isz(n) and not any(isz(d) for d in divisors(n)[:-1])
print(list(filter(ok, range(1, 5000)))) # Michael S. Branicky, Jun 20 2021

SageMath

# uses[is_Zumkeller from A083207]
def is_primitiveZumkeller(n):
    return (is_Zumkeller(n) and
        not any(is_Zumkeller(d) for d in divisors(n)[:-1]))
print([n for n in (1..4216) if is_primitiveZumkeller(n)]) # Peter Luschny, Jun 21 2021